Time Rates | Applications

Solving Time Rates by Chain Rule | Differential Calculus

Time Rates
If a quantity x is a function of time t, the time rate of change of x is given by dx/dt.

When two or more quantities, all functions of t, are related by an equation, the relation between their rates of change may be obtained by differentiating both sides of the equation with respect to t.

Basic Time Rates

  • Velocity, $v = \dfrac{ds}{dt}$, where $s$ is the distance.
  • Acceleration, $a = \dfrac{dv}{dt} = \dfrac{d^2s}{dt^2}$, where $v$ is velocity and $s$ is the distance.
  • Discharge, $Q = \dfrac{dV}{dt}$, where $V$ is the volume at any time.
  • Angular Speed, $\omega = \dfrac{d\theta}{dt}$, where $\theta$ is the angle at any time.


Steps in Solving Time Rates Problem

  1. Identify what are changing and what are fixed.
  2. Assign variables to those that are changing and appropriate value (constant) to those that are fixed.
  3. Create an equation relating all the variables and constants in Step 2.
  4. Differentiate the equation with respect to time.